L(s) = 1 | + i·2-s + (0.645 + 1.60i)3-s − 4-s + (0.866 + 0.5i)5-s + (−1.60 + 0.645i)6-s + (−2.35 − 4.08i)7-s − i·8-s + (−2.16 + 2.07i)9-s + (−0.5 + 0.866i)10-s + (−3.18 + 5.51i)11-s + (−0.645 − 1.60i)12-s + (−2.96 − 1.71i)13-s + (4.08 − 2.35i)14-s + (−0.244 + 1.71i)15-s + 16-s + (−1.65 − 2.87i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.372 + 0.928i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.656 + 0.263i)6-s + (−0.890 − 1.54i)7-s − 0.353i·8-s + (−0.722 + 0.691i)9-s + (−0.158 + 0.273i)10-s + (−0.960 + 1.66i)11-s + (−0.186 − 0.464i)12-s + (−0.823 − 0.475i)13-s + (1.09 − 0.629i)14-s + (−0.0632 + 0.442i)15-s + 0.250·16-s + (−0.402 − 0.696i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0672838 - 0.0843331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0672838 - 0.0843331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.645 - 1.60i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + (-3.67 - 4.18i)T \) |
good | 7 | \( 1 + (2.35 + 4.08i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.18 - 5.51i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.96 + 1.71i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.65 + 2.87i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.56 + 4.43i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 37 | \( 1 + (4.26 - 2.46i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.33 + 4.81i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.26 - 3.04i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.50iT - 47T^{2} \) |
| 53 | \( 1 + (0.645 - 1.11i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.87 + 2.81i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 1.31iT - 61T^{2} \) |
| 67 | \( 1 + (-7.24 + 12.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.61 - 0.929i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-9.05 - 5.22i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.954 + 0.551i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.19 - 5.54i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6.30T + 89T^{2} \) |
| 97 | \( 1 - 0.469T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32280235316707280035178739144, −9.822563666513387420208602542377, −9.254461113079039227576854223053, −8.009095289811465092994346946776, −7.08847555323134951852891823636, −6.79714046628810875311486430465, −5.00438238468169601223925932878, −4.83272933206604909315698521438, −3.57149120873460665224511552075, −2.51662744453653316119130503386,
0.04387656869901756866960531454, 1.91713561669057646016777634617, 2.69523306805172139117949039301, 3.52054410769315308921300217652, 5.36188240425726337677262054797, 5.87601636584778677934484112540, 6.76872419500676067383085513695, 8.260863469818985983270459003080, 8.578934663852447789978065286094, 9.361688313456697793767184229994